All Questions
58
questions
4
votes
1
answer
161
views
Counterexample for a proof
Let $n$ and $k$ be positive integers and
$$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$
be a lattice cube of length $n$.
Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red ...
- 401
3
votes
2
answers
97
views
Counting $10$ length paths in a $2 \times 4$ rectangle with distance $6$ units from start to end meaning negative moves allowed?
How many different routes of length 10 units (each side is 1 unit) are there to traverse from lower left corner (point A) to top right corner (point B) in a rectangle with 2 rows and 4 column cells ...
- 125
5
votes
1
answer
245
views
Taking stones game beginning with 1 to 4 stones in a 2 player game. If we started with 18 stones, is the a winning strategy for the first player?
Amy and Beck are playing 'taking the stones game'. There are 18 stones on the table, and the two people take stones in turns. The first move of the starting player can take 1 to 4 stones. For the ...
- 125
6
votes
1
answer
438
views
Placing the $21$ two-digit primes into a grid, such that primes in adjacent squares have either the same tens digit or ones digit
This is USAMTS round 3, problem 1 of the 2020-2021 Academic Year.
Place the 21 2-digit prime numbers in the white squares of the grid on the right so that each two-digit prime is used exactly once. ...
- 1,837
3
votes
1
answer
83
views
Find max N people with 2022 different colored balls choosing two balls each so that no 3 people exactly contain balls of 3 colours
Here is the "exact" wording given in the problem.
There are $N$ people. Each person gets two balls of different colours among $2022$ balls with different colours. The combinations of the ...
- 125
2
votes
1
answer
102
views
Find the number of subsets of n chairs in a circle containing at least three adjacent chairs
Find the number of subsets of $n$ chairs in a circle containing at least three adjacent chairs.
I know that the answer for $n=10$ is $581$, and the solution is here for instance.
I'm not sure if it's ...
- 469
1
vote
1
answer
73
views
How many good distributions of coins are there?
At each vertex in figure 1, there is a student. A total of n = 60k coins are distributed to these students. The coins are redistributed as follows: Each student simultaneously give an equal number of ...
- 469
1
vote
1
answer
140
views
Difficulty with a Combinatorial Math problem
I was doing this question from a national mathematical olympiad and although I couldn't solve it but I found something but don't know how to progress.
For a positive integer $N$, let $T(N)$ denote the ...
- 131
0
votes
0
answers
233
views
Can we arrange $\{1,...,9\}$ in $3\times 3$ grid so the set of products of rows equals the set of products of columns? [duplicate]
I find a interesting question of Prmo mock and Promys 2020
For which $n\in\mathbb{N}$ is it possible to arrange $\{1,…,n^2\}$ in an $n\times n$ grid so that the set of products of columns equals the ...
user1147681
4
votes
4
answers
375
views
What is the probability that the sum of 6 four-sided dice is less than or equal to 14?
The solution given is shown below.
My question is how did they count the numerator like that?
What is the explanation for it please?
$$\begin{align}
\frac{C_6^{14}-C_1^6\times C_6^{10}+C_2^6 \times ...
2
votes
1
answer
89
views
With sachets containing at most 13 seeds, given 2021 rose, jasmine or fennel seeds respectively what is the maximum number of rose seeds used?
I have taken a part of the problem which I did not understand its solution.
Basically, there are:
$2021$ rose seeds,
$2021$ jasmine seeds and
$2021$ fennel seeds.
So now, with sachets of at most size ...
4
votes
3
answers
272
views
How many ways for a beetle to move from bottom left corner to upper right corner in a 6 x 6 grid if it must be done in 14 steps only?
Note: We allow all four directions (up, down, right, left but no diagonal)
The 6 x 6 grid is composed of 7 horizontal lines and 7 vertical lines. We are to calculate how many 14 steps paths are ...
0
votes
1
answer
71
views
find all good numbers between 500 and 528
For which positive integers $500\le n\le 528$ does there exist a positive integer k so that the numbers from 1 to 3k can be split into k pairwise disjoint subsets, each of size 3 and each of which ...
- 2,031
1
vote
1
answer
128
views
prove that the number of students in the school is at most k
Students in a school go for ice cream in groups of at least two. After $k > 1$ groups have gone, every two students have gone together exactly once. Prove that the number of students in the school ...
- 1,837
1
vote
1
answer
106
views
how many white squares does a $2022 \times 2021$ block have?
An infinite checkerboard is coloured black and white so that every $2\times 3$ block has exactly two white squares. Prove (or disprove) that every $2022\times 2021$ block has the same number of white ...
- 2,031